In 1949, Andre Weil published striking conjectures linking number theory to topology and a striking strategy for a proof [Weil, 1949]. Around 1953, Jean-Pierre Serre took on the project and soon recruited Alexander Grothendieck. Serre created aseries of concise elegant tools which Grothendieck and coworkers simplified into thousands of pages of category theory. Some have complained of this styl...

In SGA 2, Grothendieck conjectures that the étale fundamental group of punctured spectrum a complete noetherian local domain dimension at least two with algebraically closed residue field is topologically finitely generated. this paper, we prove weaker statement, namely maximal pro-nilpotent quotient The proof uses p-adic nearby cycles and negative definiteness intersection pairings over resolu...

We give a nonrecursive combinatorial formula for the expansion of a stable Grothendieck polynomial in the basis of stable Grothendieck polynomials for partitions. The proof is based on a generalization of the EdelmanGreene insertion algorithm. This result is applied to prove a number of formulas and properties for K-theoretic quiver polynomials and Grothendieck polynomials. In particular we for...

The little Grothendieck problem consists of maximizing Σ ij Cijxixj for a positive semidef-inite matrix C, over binary variables xi ∈ {±1}. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C ∈ ℝ dn × dn a positive semidefinite matrix, the objective is to maximize [Formula: see text] restricting Oi to take values...

We prove the triviality of the Grothendieck ring of a Z-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K to itself minus a point. When we specialize to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point. At the Edinburgh meeting on the model theory ...

A combinatorial expansion of the Hall-Littlewood functions into the Schur basis of symmetric functions was first given by Lascoux and Schützenberger, with their discovery of the charge statistic. A combinatorial expansion of stable Grassmannian Grothendieck polynomials into monomials was first given by Buch, using set-valued tableaux. The dual basis of the stable Grothendieck polynomials was gi...

Motivated by the study of Gabriel dimension of a Grothendieck category, we introduce the concept of atomical Grothendieck category, which has only two localizing subcategories, and we give a classification of this type of Grothendieck categories. 1. Introduction. Given a Grothendieck category Ꮽ, we can associate with it the lattice of all localizing categories of Ꮽ and denote it by Tors(Ꮽ). We ...

Quantum K-theory is a K-theoretic version of quantum cohomology, which was recently defined by Y.-P. Lee. Based on a presentation for the quantum K-theory of the classical flag variety Fln, we define and study quantum Grothendieck polynomials. We conjecture that they represent Schubert classes (i.e., the natural basis elements) in the quantum K-theory of Fln, and present strong evidence for thi...